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搜索</span></a></div><div class="menus_items"><div class="menus_item"><a class="site-page" href="/"><i class="fa-fw fas fa-home"></i><span> 主页</span></a></div><div class="menus_item"><a class="site-page group hide" href="javascript:void(0);"><i class="fa-fw fa fa-bookmark"></i><span> 笔记目录</span><i class="fas fa-chevron-down"></i></a><ul class="menus_item_child"><li><a class="site-page child" href="/categories/"><i class="fa-fw fa fa-folder-open"></i><span> 分类</span></a></li><li><a class="site-page child" href="/tags/"><i class="fa-fw fa fa-tags"></i><span> 标签</span></a></li><li><a class="site-page child" href="/archives/"><i class="fa-fw fa fa-archive"></i><span> 归档</span></a></li></ul></div><div class="menus_item"><a class="site-page group hide" href="javascript:void(0);"><i class="fa-fw fa fa-blog"></i><span> 其他博客</span><i class="fas fa-chevron-down"></i></a><ul class="menus_item_child"><li><a class="site-page child" target="_blank" rel="noopener" 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href="javascript:void(0);"><i class="fas fa-bars fa-fw"></i></a></div></div></nav><div id="post-info"><h1 class="post-title">二叉树和线索二叉树的C++语言描述实现模板</h1><div id="post-meta"><div class="meta-firstline"><span class="post-meta-date"><i class="far fa-calendar-alt fa-fw post-meta-icon"></i><span class="post-meta-label">发表于</span><time class="post-meta-date-created" datetime="2022-08-31T04:18:34.000Z" title="发表于 2022-08-31 12:18:34">2022-08-31</time><span class="post-meta-separator">|</span><i class="fas fa-history fa-fw post-meta-icon"></i><span class="post-meta-label">更新于</span><time class="post-meta-date-updated" datetime="2023-04-13T03:12:17.109Z" title="更新于 2023-04-13 11:12:17">2023-04-13</time></span><span class="post-meta-categories"><span class="post-meta-separator">|</span><i class="fas fa-inbox fa-fw post-meta-icon"></i><a class="post-meta-categories" href="/categories/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/">数据结构</a></span></div><div class="meta-secondline"><span class="post-meta-separator">|</span><span class="post-meta-pv-cv" id="" data-flag-title="二叉树和线索二叉树的C++语言描述实现模板"><i class="far fa-eye fa-fw post-meta-icon"></i><span class="post-meta-label">阅读量:</span><span id="busuanzi_value_page_pv"><i class="fa-solid fa-spinner fa-spin"></i></span></span></div></div></div></header><main class="layout" id="content-inner"><div id="post"><article class="post-content" id="article-container"><h1 id="前言"><a href="#前言" class="headerlink" title="前言"></a>前言</h1><p>数据结构中二叉树和线索二叉树的C++语言描述实现模板，有详细的步骤解析及使用示例。</p>
<hr>
<h1 id="代码仓库"><a href="#代码仓库" class="headerlink" title="代码仓库"></a>代码仓库</h1><ul>
<li><a target="_blank" rel="noopener" href="https://github.com/yezhening/Programming-examples">yezhening&#x2F;Programming-examples: 编程实例 (github.com)</a></li>
<li><a target="_blank" rel="noopener" href="https://gitee.com/yezhening/Programming-examples">Programming-examples: 编程实例 (gitee.com)</a></li>
</ul>
<hr>
<h1 id="bTree-cpp（二叉树）"><a href="#bTree-cpp（二叉树）" class="headerlink" title="bTree.cpp（二叉树）"></a>bTree.cpp（二叉树）</h1><figure class="highlight cpp"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span 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class="line">208</span><br><span class="line">209</span><br><span class="line">210</span><br><span class="line">211</span><br><span class="line">212</span><br><span class="line">213</span><br><span class="line">214</span><br><span class="line">215</span><br><span class="line">216</span><br><span class="line">217</span><br><span class="line">218</span><br><span class="line">219</span><br><span class="line">220</span><br><span class="line">221</span><br><span class="line">222</span><br><span class="line">223</span><br><span class="line">224</span><br><span class="line">225</span><br><span class="line">226</span><br><span class="line">227</span><br><span class="line">228</span><br><span class="line">229</span><br><span class="line">230</span><br><span class="line">231</span><br><span class="line">232</span><br><span class="line">233</span><br><span class="line">234</span><br><span class="line">235</span><br><span class="line">236</span><br><span class="line">237</span><br><span class="line">238</span><br><span class="line">239</span><br><span class="line">240</span><br><span class="line">241</span><br><span class="line">242</span><br><span class="line">243</span><br><span class="line">244</span><br><span class="line">245</span><br><span class="line">246</span><br><span class="line">247</span><br><span class="line">248</span><br><span class="line">249</span><br><span class="line">250</span><br><span class="line">251</span><br><span class="line">252</span><br><span class="line">253</span><br><span class="line">254</span><br><span class="line">255</span><br><span class="line">256</span><br><span class="line">257</span><br><span class="line">258</span><br><span class="line">259</span><br><span class="line">260</span><br><span class="line">261</span><br><span class="line">262</span><br><span class="line">263</span><br><span class="line">264</span><br><span class="line">265</span><br><span class="line">266</span><br><span class="line">267</span><br><span class="line">268</span><br><span class="line">269</span><br><span class="line">270</span><br><span class="line">271</span><br><span class="line">272</span><br><span class="line">273</span><br><span class="line">274</span><br><span class="line">275</span><br><span class="line">276</span><br><span class="line">277</span><br><span class="line">278</span><br><span class="line">279</span><br><span class="line">280</span><br><span class="line">281</span><br><span class="line">282</span><br><span class="line">283</span><br><span class="line">284</span><br><span class="line">285</span><br><span class="line">286</span><br><span class="line">287</span><br><span class="line">288</span><br><span class="line">289</span><br><span class="line">290</span><br><span class="line">291</span><br><span class="line">292</span><br><span class="line">293</span><br><span class="line">294</span><br><span class="line">295</span><br><span class="line">296</span><br><span class="line">297</span><br><span class="line">298</span><br><span class="line">299</span><br><span class="line">300</span><br><span class="line">301</span><br><span class="line">302</span><br><span class="line">303</span><br><span class="line">304</span><br><span class="line">305</span><br><span class="line">306</span><br><span class="line">307</span><br><span class="line">308</span><br><span class="line">309</span><br><span class="line">310</span><br><span class="line">311</span><br><span class="line">312</span><br><span class="line">313</span><br><span class="line">314</span><br><span class="line">315</span><br><span class="line">316</span><br><span class="line">317</span><br><span class="line">318</span><br><span class="line">319</span><br><span class="line">320</span><br><span class="line">321</span><br><span class="line">322</span><br><span class="line">323</span><br><span class="line">324</span><br><span class="line">325</span><br><span class="line">326</span><br><span class="line">327</span><br><span class="line">328</span><br><span class="line">329</span><br><span class="line">330</span><br><span class="line">331</span><br><span class="line">332</span><br><span class="line">333</span><br><span class="line">334</span><br><span class="line">335</span><br><span class="line">336</span><br><span class="line">337</span><br><span class="line">338</span><br><span class="line">339</span><br><span class="line">340</span><br><span class="line">341</span><br><span class="line">342</span><br><span class="line">343</span><br><span class="line">344</span><br><span class="line">345</span><br><span class="line">346</span><br><span class="line">347</span><br><span class="line">348</span><br><span class="line">349</span><br><span class="line">350</span><br><span class="line">351</span><br><span class="line">352</span><br><span class="line">353</span><br><span class="line">354</span><br><span class="line">355</span><br><span class="line">356</span><br><span class="line">357</span><br><span class="line">358</span><br><span class="line">359</span><br><span class="line">360</span><br><span class="line">361</span><br><span class="line">362</span><br><span class="line">363</span><br><span class="line">364</span><br><span class="line">365</span><br><span class="line">366</span><br><span class="line">367</span><br><span class="line">368</span><br><span class="line">369</span><br><span class="line">370</span><br><span class="line">371</span><br><span class="line">372</span><br><span class="line">373</span><br><span class="line">374</span><br><span class="line">375</span><br><span class="line">376</span><br><span class="line">377</span><br><span class="line">378</span><br><span class="line">379</span><br><span class="line">380</span><br><span class="line">381</span><br><span class="line">382</span><br><span class="line">383</span><br><span class="line">384</span><br><span class="line">385</span><br><span class="line">386</span><br><span class="line">387</span><br><span class="line">388</span><br><span class="line">389</span><br><span class="line">390</span><br><span class="line">391</span><br><span class="line">392</span><br><span class="line">393</span><br><span class="line">394</span><br><span class="line">395</span><br><span class="line">396</span><br><span class="line">397</span><br><span class="line">398</span><br><span class="line">399</span><br><span class="line">400</span><br><span class="line">401</span><br><span class="line">402</span><br><span class="line">403</span><br><span class="line">404</span><br><span class="line">405</span><br><span class="line">406</span><br><span class="line">407</span><br><span class="line">408</span><br><span class="line">409</span><br><span class="line">410</span><br><span class="line">411</span><br><span class="line">412</span><br><span class="line">413</span><br><span class="line">414</span><br><span class="line">415</span><br><span class="line">416</span><br><span class="line">417</span><br><span class="line">418</span><br><span class="line">419</span><br><span class="line">420</span><br><span class="line">421</span><br><span class="line">422</span><br><span class="line">423</span><br><span class="line">424</span><br><span class="line">425</span><br><span class="line">426</span><br><span class="line">427</span><br><span class="line">428</span><br><span class="line">429</span><br><span class="line">430</span><br><span class="line">431</span><br><span class="line">432</span><br><span class="line">433</span><br><span class="line">434</span><br><span class="line">435</span><br><span class="line">436</span><br><span class="line">437</span><br><span class="line">438</span><br><span class="line">439</span><br><span class="line">440</span><br><span class="line">441</span><br><span class="line">442</span><br><span class="line">443</span><br><span class="line">444</span><br><span class="line">445</span><br><span class="line">446</span><br><span class="line">447</span><br><span class="line">448</span><br><span class="line">449</span><br><span class="line">450</span><br><span class="line">451</span><br><span class="line">452</span><br><span class="line">453</span><br><span class="line">454</span><br><span class="line">455</span><br><span class="line">456</span><br><span class="line">457</span><br><span class="line">458</span><br><span class="line">459</span><br><span class="line">460</span><br><span class="line">461</span><br><span class="line">462</span><br><span class="line">463</span><br><span class="line">464</span><br><span class="line">465</span><br><span class="line">466</span><br><span class="line">467</span><br><span class="line">468</span><br><span class="line">469</span><br><span class="line">470</span><br><span class="line">471</span><br><span class="line">472</span><br><span class="line">473</span><br><span class="line">474</span><br><span class="line">475</span><br></pre></td><td class="code"><pre><span class="line"><span class="comment">//头文件————————————————————</span></span><br><span class="line"><span class="meta">#<span class="keyword">include</span> <span class="string">&lt;iostream&gt;</span></span></span><br><span class="line"><span class="meta">#<span class="keyword">include</span> <span class="string">&lt;vector&gt;</span></span></span><br><span class="line"></span><br><span class="line"><span class="comment">//命名空间</span></span><br><span class="line"><span class="keyword">using</span> <span class="keyword">namespace</span> std;</span><br><span class="line"></span><br><span class="line"><span class="comment">//自定义数据类型————————————————————</span></span><br><span class="line"><span class="keyword">typedef</span> <span class="type">int</span> ELEM_TYPE; <span class="comment">//数据的（数据）类型 依据数据的实际类型定义</span></span><br><span class="line"></span><br><span class="line"><span class="comment">//结构体</span></span><br><span class="line"><span class="comment">//定义</span></span><br><span class="line"><span class="comment">//结点</span></span><br><span class="line"><span class="keyword">typedef</span> <span class="keyword">struct</span> <span class="title class_">BTNode</span></span><br><span class="line">&#123;</span><br><span class="line">    ELEM_TYPE data;        <span class="comment">//数据</span></span><br><span class="line">    <span class="keyword">struct</span> <span class="title class_">BTNode</span> *lChild; <span class="comment">//指向左孩子结点的指针</span></span><br><span class="line">    <span class="keyword">struct</span> <span class="title class_">BTNode</span> *rChild; <span class="comment">//指向右孩子结点的指针</span></span><br><span class="line">&#125; BTNode;</span><br><span class="line"><span class="comment">// struct BTNode：结构体数据类型的名称</span></span><br><span class="line"><span class="comment">//如创建一个结点的语句：struct BTNode node;</span></span><br><span class="line"><span class="comment">//若结构体数据类型内，存在指向该结构体的指针数据类型，则必须完整命名结构体数据类型的名称</span></span><br><span class="line"><span class="comment">//如结点结构体BTNode内，存在指向该结构体的指针数据类型lChild和rChild，则命名语句为：typedef struct BTNode而不是typedef struct</span></span><br><span class="line"><span class="comment">// &#125;BTNode;：typedef给结构体数据类型起的别名，可简化语句</span></span><br><span class="line"><span class="comment">//如创建一个结点的语句：BTNode node;</span></span><br><span class="line"><span class="comment">//结构体数据类型的名称和别名尽量一致，以方便记忆、使用</span></span><br><span class="line"></span><br><span class="line"><span class="comment">//函数声明————————————————————</span></span><br><span class="line"><span class="function"><span class="type">void</span> <span class="title">use_example</span><span class="params">()</span></span>;                     <span class="comment">//使用示例</span></span><br><span class="line"><span class="function"><span class="type">void</span> <span class="title">createBTree</span><span class="params">(<span class="keyword">struct</span> BTNode *&amp;root)</span></span>; <span class="comment">//创建 前序法</span></span><br><span class="line"></span><br><span class="line"><span class="function"><span class="type">void</span> <span class="title">preOrderTraverse</span><span class="params">(<span class="keyword">struct</span> BTNode *root)</span></span>;                                              <span class="comment">//前序遍历    递归法</span></span><br><span class="line"><span class="function"><span class="type">void</span> <span class="title">inOrderTraverse</span><span class="params">(<span class="keyword">struct</span> BTNode *root)</span></span>;                                               <span class="comment">//中序遍历    递归法</span></span><br><span class="line"><span class="function"><span class="type">void</span> <span class="title">postOrderTraverse</span><span class="params">(<span class="keyword">struct</span> BTNode *root)</span></span>;                                             <span class="comment">//后序遍历    递归法</span></span><br><span class="line"><span class="function"><span class="type">void</span> <span class="title">levelOrderTraverse</span><span class="params">(<span class="keyword">struct</span> BTNode *root, <span class="type">int</span> layer, vector&lt;vector&lt;ELEM_TYPE&gt;&gt; &amp;res)</span></span>; <span class="comment">//层序遍历   递归法</span></span><br><span class="line"></span><br><span class="line"><span class="function"><span class="type">void</span> <span class="title">preOrderTraverse2</span><span class="params">(<span class="keyword">struct</span> BTNode *root)</span></span>;   <span class="comment">//前序遍历  迭代法</span></span><br><span class="line"><span class="function"><span class="type">void</span> <span class="title">inOrderTraverse2</span><span class="params">(<span class="keyword">struct</span> BTNode *root)</span></span>;    <span class="comment">//中序遍历  迭代法</span></span><br><span class="line"><span class="function"><span class="type">void</span> <span class="title">postOrderTraverse2</span><span class="params">(<span class="keyword">struct</span> BTNode *root)</span></span>;  <span class="comment">//后序遍历  迭代法</span></span><br><span class="line"><span class="function"><span class="type">void</span> <span class="title">levelOrderTraverse2</span><span class="params">(<span class="keyword">struct</span> BTNode *root)</span></span>; <span class="comment">//层序遍历   迭代法</span></span><br><span class="line"></span><br><span class="line"><span class="function"><span class="type">int</span> <span class="title">getDepth</span><span class="params">(<span class="keyword">struct</span> BTNode *root)</span></span>; <span class="comment">//获取深度/高度</span></span><br><span class="line"></span><br><span class="line"><span class="comment">//函数定义————————————————————</span></span><br><span class="line"><span class="comment">//使用示例</span></span><br><span class="line"><span class="function"><span class="type">void</span> <span class="title">use_example</span><span class="params">()</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="comment">//创建</span></span><br><span class="line">    <span class="keyword">struct</span> <span class="title class_">BTNode</span> *root; <span class="comment">//根结点指针</span></span><br><span class="line"></span><br><span class="line">    <span class="built_in">createBTree</span>(root); <span class="comment">//创建 前序法</span></span><br><span class="line">    <span class="comment">//构造三层二叉树：</span></span><br><span class="line">    <span class="comment">//第一层：1</span></span><br><span class="line">    <span class="comment">//第二层：2，3</span></span><br><span class="line">    <span class="comment">//第三层：-1，-1，-1，4</span></span><br><span class="line">    <span class="comment">//输入数据：1，2，-1，-1，3，-1，4，-1，-1</span></span><br><span class="line"></span><br><span class="line">    <span class="built_in">preOrderTraverse</span>(root); <span class="comment">//前序遍历  递归法   输出：1，2，3，4</span></span><br><span class="line">    cout &lt;&lt; endl;</span><br><span class="line"></span><br><span class="line">    <span class="built_in">inOrderTraverse</span>(root); <span class="comment">//中序遍历   递归法   输出：2，1，3，4</span></span><br><span class="line">    cout &lt;&lt; endl;</span><br><span class="line"></span><br><span class="line">    <span class="built_in">postOrderTraverse</span>(root); <span class="comment">//后序遍历 递归法    输出：2，4，3，1</span></span><br><span class="line">    cout &lt;&lt; endl;</span><br><span class="line"></span><br><span class="line">    vector&lt;vector&lt;ELEM_TYPE&gt;&gt; res; <span class="comment">//层序遍历 递归法 输出：1，2，3，4</span></span><br><span class="line">    <span class="built_in">levelOrderTraverse</span>(root, <span class="number">0</span>, res);</span><br><span class="line">    <span class="keyword">for</span> (vector&lt;ELEM_TYPE&gt; vec : res)</span><br><span class="line">    &#123;</span><br><span class="line">        <span class="keyword">for</span> (ELEM_TYPE data : vec)</span><br><span class="line">        &#123;</span><br><span class="line">            cout &lt;&lt; data;</span><br><span class="line">        &#125;</span><br><span class="line">    &#125;</span><br><span class="line">    cout &lt;&lt; endl;</span><br><span class="line"></span><br><span class="line">    <span class="built_in">preOrderTraverse2</span>(root); <span class="comment">//前序遍历  迭代法   输出：1，2，3，4</span></span><br><span class="line">    cout &lt;&lt; endl;</span><br><span class="line"></span><br><span class="line">    <span class="built_in">inOrderTraverse2</span>(root); <span class="comment">//中序遍历  迭代法    输出：2，1，3，4</span></span><br><span class="line">    cout &lt;&lt; endl;</span><br><span class="line"></span><br><span class="line">    <span class="built_in">postOrderTraverse2</span>(root); <span class="comment">//后序遍历  迭代法    输出：2，4，3，1</span></span><br><span class="line">    cout &lt;&lt; endl;</span><br><span class="line"></span><br><span class="line">    <span class="built_in">levelOrderTraverse2</span>(root); <span class="comment">//层序遍历 迭代法 输出：1，2，3，4</span></span><br><span class="line">    cout &lt;&lt; endl;</span><br><span class="line"></span><br><span class="line">    cout &lt;&lt; <span class="built_in">getDepth</span>(root) &lt;&lt; endl; <span class="comment">//获取深度/高度 输出：3</span></span><br><span class="line"></span><br><span class="line">    <span class="keyword">return</span>;</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="comment">//创建 前序法</span></span><br><span class="line"><span class="comment">//时间复杂度：O(n)。n为数据规模</span></span><br><span class="line"><span class="comment">//空间复杂度：O(n)。n为数据规模</span></span><br><span class="line"><span class="comment">//空间复杂度：递归调用栈的规模/树的深度/高度：</span></span><br><span class="line"><span class="comment">//最好：[log以2为底的n]下取整+1或[log以2为底的(n+1)]上取整。n为数据规模</span></span><br><span class="line"><span class="comment">//最坏：n。n为数据规模</span></span><br><span class="line"><span class="comment">//注意：参数使用*&amp;，指向指针的引用数据类型</span></span><br><span class="line"><span class="comment">//若只使用*：指针数据类型，在函数体中并未进行解引用调用修改*root的值，使用malloc()修改的是root的值</span></span><br><span class="line"><span class="comment">//所以是值拷贝，函数返回后数据丢失。地址拷贝需要引用数据类型的配合</span></span><br><span class="line"><span class="function"><span class="type">void</span> <span class="title">createBTree</span><span class="params">(<span class="keyword">struct</span> BTNode *&amp;root)</span> <span class="comment">//参数：根结点指针</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    ELEM_TYPE value = <span class="number">0</span>; <span class="comment">//值</span></span><br><span class="line">    cin &gt;&gt; value;        <span class="comment">//获取输入值</span></span><br><span class="line"></span><br><span class="line">    <span class="keyword">if</span> (value == <span class="number">-1</span>) <span class="comment">//空树  约定值为-1时无结点</span></span><br><span class="line">    &#123;</span><br><span class="line">        root = <span class="literal">nullptr</span>; <span class="comment">//根结点指针指向空</span></span><br><span class="line">    &#125;</span><br><span class="line">    <span class="keyword">else</span> <span class="comment">//非空树</span></span><br><span class="line">    &#123;</span><br><span class="line">        root = (<span class="keyword">struct</span> BTNode *)<span class="built_in">malloc</span>(<span class="built_in">sizeof</span>(<span class="keyword">struct</span> BTNode)); <span class="comment">//创建根结点，根结点指针指向根结点</span></span><br><span class="line"></span><br><span class="line">        root-&gt;data = value;        <span class="comment">//初始化根结点的数据域</span></span><br><span class="line">        <span class="built_in">createBTree</span>(root-&gt;lChild); <span class="comment">//递归构造左子树</span></span><br><span class="line">        <span class="built_in">createBTree</span>(root-&gt;rChild); <span class="comment">//递归构造右子树</span></span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line">    <span class="keyword">return</span>;</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="comment">//前序遍历  递归法</span></span><br><span class="line"><span class="comment">//时间复杂度：O(n)。n为数据规模</span></span><br><span class="line"><span class="comment">//空间复杂度：O(n)。n为数据规模</span></span><br><span class="line"><span class="comment">//空间复杂度：递归调用栈的规模/树的深度/高度：</span></span><br><span class="line"><span class="comment">//最好：[log以2为底的n]下取整+1或[log以2为底的(n+1)]上取整。n为数据规模</span></span><br><span class="line"><span class="comment">//最坏：n。n为数据规模</span></span><br><span class="line"><span class="function"><span class="type">void</span> <span class="title">preOrderTraverse</span><span class="params">(<span class="keyword">struct</span> BTNode *root)</span> <span class="comment">//参数：根结点指针</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="keyword">if</span> (root == <span class="literal">nullptr</span>) <span class="comment">//空树</span></span><br><span class="line">    &#123;</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="keyword">else</span> <span class="comment">//非空树</span></span><br><span class="line">    &#123;</span><br><span class="line">        cout &lt;&lt; root-&gt;data;             <span class="comment">//输出数据</span></span><br><span class="line">        <span class="built_in">preOrderTraverse</span>(root-&gt;lChild); <span class="comment">//前序遍历左子树</span></span><br><span class="line">        <span class="built_in">preOrderTraverse</span>(root-&gt;rChild); <span class="comment">//前序遍历右子树</span></span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line">    <span class="keyword">return</span>;</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="comment">//中序遍历  递归法</span></span><br><span class="line"><span class="comment">//时间复杂度：O(n)。n为数据规模</span></span><br><span class="line"><span class="comment">//空间复杂度：O(n)。n为数据规模</span></span><br><span class="line"><span class="comment">//空间复杂度：递归调用栈的规模/树的深度/高度：</span></span><br><span class="line"><span class="comment">//最好：[log以2为底的n]下取整+1或[log以2为底的(n+1)]上取整。n为数据规模</span></span><br><span class="line"><span class="comment">//最坏：n。n为数据规模</span></span><br><span class="line"><span class="function"><span class="type">void</span> <span class="title">inOrderTraverse</span><span class="params">(<span class="keyword">struct</span> BTNode *root)</span> <span class="comment">//参数：根结点指针</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="keyword">if</span> (root == <span class="literal">nullptr</span>) <span class="comment">//空树</span></span><br><span class="line">    &#123;</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="keyword">else</span> <span class="comment">//非空树</span></span><br><span class="line">    &#123;</span><br><span class="line">        <span class="built_in">inOrderTraverse</span>(root-&gt;lChild); <span class="comment">//中序遍历左子树</span></span><br><span class="line">        cout &lt;&lt; root-&gt;data;            <span class="comment">//输出数据</span></span><br><span class="line">        <span class="built_in">inOrderTraverse</span>(root-&gt;rChild); <span class="comment">//中序遍历右子树</span></span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line">    <span class="keyword">return</span>;</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="comment">//后序遍历  递归法</span></span><br><span class="line"><span class="comment">//时间复杂度：O(n)。n为数据规模</span></span><br><span class="line"><span class="comment">//空间复杂度：O(n)。n为数据规模</span></span><br><span class="line"><span class="comment">//空间复杂度：递归调用栈的规模/树的深度/高度：</span></span><br><span class="line"><span class="comment">//最好：[log以2为底的n]下取整+1或[log以2为底的(n+1)]上取整。n为数据规模</span></span><br><span class="line"><span class="comment">//最坏：n。n为数据规模</span></span><br><span class="line"><span class="function"><span class="type">void</span> <span class="title">postOrderTraverse</span><span class="params">(<span class="keyword">struct</span> BTNode *root)</span> <span class="comment">//参数：根结点指针</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="keyword">if</span> (root == <span class="literal">nullptr</span>) <span class="comment">//空树</span></span><br><span class="line">    &#123;</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="keyword">else</span> <span class="comment">//非空树</span></span><br><span class="line">    &#123;</span><br><span class="line">        <span class="built_in">postOrderTraverse</span>(root-&gt;lChild); <span class="comment">//后序遍历左子树</span></span><br><span class="line">        <span class="built_in">postOrderTraverse</span>(root-&gt;rChild); <span class="comment">//后序遍历右子树</span></span><br><span class="line">        cout &lt;&lt; root-&gt;data;              <span class="comment">//输出数据</span></span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line">    <span class="keyword">return</span>;</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="comment">//层序遍历  递归法</span></span><br><span class="line"><span class="comment">//时间复杂度：O(n)。n为数据规模</span></span><br><span class="line"><span class="comment">//空间复杂度：O(n)。n为数据规模</span></span><br><span class="line"><span class="comment">//空间复杂度：递归调用栈的规模/树的深度/高度：</span></span><br><span class="line"><span class="comment">//最好：[log以2为底的n]下取整+1或[log以2为底的(n+1)]上取整。n为数据规模</span></span><br><span class="line"><span class="comment">//最坏：n。n为数据规模</span></span><br><span class="line"><span class="comment">//空间复杂度：存储结点数据的二维向量：n。n为数据规模</span></span><br><span class="line"><span class="function"><span class="type">void</span> <span class="title">levelOrderTraverse</span><span class="params">(<span class="keyword">struct</span> BTNode *root, <span class="type">int</span> layer, vector&lt;vector&lt;ELEM_TYPE&gt;&gt; &amp;res)</span> <span class="comment">//参数：根结点指针，层次，记录结点数据的二维向量</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="keyword">if</span> (root == <span class="literal">nullptr</span>) <span class="comment">//空树</span></span><br><span class="line">    &#123;</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="keyword">else</span> <span class="comment">//非空树</span></span><br><span class="line">    &#123;</span><br><span class="line">        <span class="comment">//一维向量vec记录一层中，从左到右结点的数据</span></span><br><span class="line">        <span class="comment">//二维向量res记录从上到下每一层的vec</span></span><br><span class="line"></span><br><span class="line">        <span class="comment">//因为向量大小从0开始，约定：层次从0开始，根结点为第0层，根结点的孩子结点为第1层，依次类推</span></span><br><span class="line">        <span class="comment">//每遍历到新的层次，当前层次layer等于res的大小，如：</span></span><br><span class="line">        <span class="comment">//第一层：layer为0，res.size()为0</span></span><br><span class="line">        <span class="comment">//第二层：layer为1，res已记录第0层的vec，res.size()为1</span></span><br><span class="line">        <span class="comment">//每遍历到新的层次，创建vec，res记录vec</span></span><br><span class="line"></span><br><span class="line">        <span class="comment">//依据layer定位res中的vec，记录结点的数据</span></span><br><span class="line"></span><br><span class="line">        <span class="comment">//核心思想：</span></span><br><span class="line">        <span class="comment">//本质：前序递归遍历，需要递归访问根结点的左子树和右子树</span></span><br><span class="line">        <span class="comment">//访问结点顺序：中-&gt;左-&gt;右，处理结点顺序：每层的左-&gt;右</span></span><br><span class="line">        <span class="comment">//所以需要结合vec、layer和res记录层次遍历结点的数据</span></span><br><span class="line"></span><br><span class="line">        <span class="keyword">if</span> (layer == res.<span class="built_in">size</span>())</span><br><span class="line">        &#123;</span><br><span class="line">            vector&lt;ELEM_TYPE&gt; vec;</span><br><span class="line">            res.<span class="built_in">push_back</span>(vec);</span><br><span class="line">        &#125;</span><br><span class="line"></span><br><span class="line">        res[layer].<span class="built_in">push_back</span>(root-&gt;data);</span><br><span class="line"></span><br><span class="line">        <span class="keyword">if</span> (root-&gt;lChild != <span class="literal">nullptr</span>) <span class="comment">//根结点的左孩子结点不为空</span></span><br><span class="line">        &#123;</span><br><span class="line">            <span class="built_in">levelOrderTraverse</span>(root-&gt;lChild, layer + <span class="number">1</span>, res);</span><br><span class="line">        &#125;</span><br><span class="line"></span><br><span class="line">        <span class="keyword">if</span> (root-&gt;rChild != <span class="literal">nullptr</span>) <span class="comment">//根结点的右孩子结点不为空</span></span><br><span class="line">        &#123;</span><br><span class="line">            <span class="built_in">levelOrderTraverse</span>(root-&gt;rChild, layer + <span class="number">1</span>, res);</span><br><span class="line">        &#125;</span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line">    <span class="keyword">return</span>;</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="comment">//前序遍历  迭代法</span></span><br><span class="line"><span class="comment">//时间复杂度：O(n)。n为数据规模</span></span><br><span class="line"><span class="comment">//空间复杂度：O(n)。n为数据规模/用户栈规模</span></span><br><span class="line"><span class="function"><span class="type">void</span> <span class="title">preOrderTraverse2</span><span class="params">(<span class="keyword">struct</span> BTNode *root)</span> <span class="comment">//参数：根结点指针</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="keyword">constexpr</span> <span class="type">int</span> maxSize = <span class="number">10</span>;             <span class="comment">//指向结点的指针的向量/栈的初始化大小    大小&gt;=结点数</span></span><br><span class="line">    <span class="function">vector&lt;<span class="keyword">struct</span> BTNode *&gt; <span class="title">stack</span><span class="params">(maxSize)</span></span>; <span class="comment">//指向结点的指针的向量/栈</span></span><br><span class="line">    <span class="type">int</span> top = <span class="number">-1</span>;                           <span class="comment">//指向栈顶结点的指针</span></span><br><span class="line"></span><br><span class="line">    <span class="keyword">struct</span> <span class="title class_">BTNode</span> *visitNode = <span class="literal">nullptr</span>; <span class="comment">//指向访问结点的指针</span></span><br><span class="line"></span><br><span class="line">    <span class="keyword">if</span> (root == <span class="literal">nullptr</span>) <span class="comment">//空树</span></span><br><span class="line">    &#123;</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="keyword">else</span> <span class="comment">//非空树</span></span><br><span class="line">    &#123;</span><br><span class="line">        ++top;</span><br><span class="line">        stack[top] = root; <span class="comment">//根结点入栈</span></span><br><span class="line"></span><br><span class="line">        <span class="keyword">while</span> (top != <span class="number">-1</span>) <span class="comment">//栈不为空时，循环</span></span><br><span class="line">        &#123;</span><br><span class="line">            visitNode = stack[top]; <span class="comment">//指向访问结点的指针指向栈顶结点</span></span><br><span class="line">            --top;                  <span class="comment">//栈顶结点出栈</span></span><br><span class="line"></span><br><span class="line">            cout &lt;&lt; visitNode-&gt;data; <span class="comment">//访问结点的数据</span></span><br><span class="line"></span><br><span class="line">            <span class="keyword">if</span> (visitNode-&gt;rChild != <span class="literal">nullptr</span>) <span class="comment">//右孩子结点不为空</span></span><br><span class="line">            &#123;</span><br><span class="line">                ++top;</span><br><span class="line">                stack[top] = visitNode-&gt;rChild; <span class="comment">//右孩子结点入栈</span></span><br><span class="line">            &#125;</span><br><span class="line"></span><br><span class="line">            <span class="keyword">if</span> (visitNode-&gt;lChild != <span class="literal">nullptr</span>) <span class="comment">//左孩子结点不为空</span></span><br><span class="line">            &#123;</span><br><span class="line">                ++top;</span><br><span class="line">                stack[top] = visitNode-&gt;lChild; <span class="comment">//左孩子结点入栈</span></span><br><span class="line">            &#125;</span><br><span class="line">        &#125;</span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line">    <span class="keyword">return</span>;</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="comment">//中序遍历  迭代法</span></span><br><span class="line"><span class="comment">//时间复杂度：O(n)。n为数据规模</span></span><br><span class="line"><span class="comment">//空间复杂度：O(n)。n为数据规模/用户栈规模</span></span><br><span class="line"><span class="function"><span class="type">void</span> <span class="title">inOrderTraverse2</span><span class="params">(<span class="keyword">struct</span> BTNode *root)</span> <span class="comment">//参数：根结点指针</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="keyword">constexpr</span> <span class="type">int</span> maxSize = <span class="number">10</span>;             <span class="comment">//指向结点的指针的向量/栈的初始化大小    大小&gt;=结点数</span></span><br><span class="line">    <span class="function">vector&lt;<span class="keyword">struct</span> BTNode *&gt; <span class="title">stack</span><span class="params">(maxSize)</span></span>; <span class="comment">//指向结点的指针的向量/栈</span></span><br><span class="line">    <span class="type">int</span> top = <span class="number">-1</span>;                           <span class="comment">//指向栈顶结点的指针</span></span><br><span class="line"></span><br><span class="line">    <span class="keyword">struct</span> <span class="title class_">BTNode</span> *visitNode = root; <span class="comment">//指向访问结点的指针    指向根结点</span></span><br><span class="line"></span><br><span class="line">    <span class="keyword">if</span> (root == <span class="literal">nullptr</span>) <span class="comment">//空树</span></span><br><span class="line">    &#123;</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="keyword">else</span> <span class="comment">//非空树</span></span><br><span class="line">    &#123;</span><br><span class="line">        <span class="comment">//注意：栈顶结点/根结点出栈并访问后，还没有访问右子树</span></span><br><span class="line">        <span class="comment">//可能存在栈空，右子树不为空的情况，需要继续遍历/循环</span></span><br><span class="line">        <span class="comment">//指向访问结点的指针指向栈顶结点/根结点的右孩子结点，若指针不空，则存在右子树，需要继续遍历/循环</span></span><br><span class="line">        <span class="keyword">while</span> (top != <span class="number">-1</span> || visitNode != <span class="literal">nullptr</span>) <span class="comment">//栈不为空或指向访问结点的指针不为空时，循环</span></span><br><span class="line">        &#123;</span><br><span class="line">            <span class="comment">//左孩子结点循环入栈</span></span><br><span class="line">            <span class="keyword">while</span> (visitNode != <span class="literal">nullptr</span>) <span class="comment">//指向访问结点的指针不为空</span></span><br><span class="line">            &#123;</span><br><span class="line">                ++top;</span><br><span class="line">                stack[top] = visitNode; <span class="comment">//指向访问结点的指针入栈</span></span><br><span class="line"></span><br><span class="line">                visitNode = visitNode-&gt;lChild; <span class="comment">//指向访问结点的指针指向访问结点的左孩子结点</span></span><br><span class="line">            &#125;</span><br><span class="line"></span><br><span class="line">            <span class="keyword">if</span> (top != <span class="number">-1</span>) <span class="comment">//栈不为空</span></span><br><span class="line">            &#123;</span><br><span class="line">                visitNode = stack[top]; <span class="comment">//指向访问结点的指针指向栈顶结点</span></span><br><span class="line">                --top;                  <span class="comment">//栈顶结点出栈</span></span><br><span class="line"></span><br><span class="line">                cout &lt;&lt; visitNode-&gt;data; <span class="comment">//访问结点的数据</span></span><br><span class="line"></span><br><span class="line">                visitNode = visitNode-&gt;rChild; <span class="comment">//指向访问结点的指针指向栈顶结点/根结点的右孩子结点</span></span><br><span class="line">            &#125;</span><br><span class="line">        &#125;</span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line">    <span class="keyword">return</span>;</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="comment">//后序遍历  迭代法</span></span><br><span class="line"><span class="comment">//时间复杂度：O(n)。n为数据规模</span></span><br><span class="line"><span class="comment">//空间复杂度：O(n)。n为数据规模/用户栈规模</span></span><br><span class="line"><span class="function"><span class="type">void</span> <span class="title">postOrderTraverse2</span><span class="params">(<span class="keyword">struct</span> BTNode *root)</span> <span class="comment">//参数：根结点指针</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="comment">//核心思想：</span></span><br><span class="line">    <span class="comment">//反前序遍历：中-&gt;右-&gt;左</span></span><br><span class="line">    <span class="comment">//逆后序遍历 = 反前序遍历</span></span><br><span class="line">    <span class="comment">//后序遍历 = 逆逆后序遍历-&gt;反前序遍历中，访问结点出栈1、入栈2再出栈2的顺序</span></span><br><span class="line"></span><br><span class="line">    <span class="keyword">constexpr</span> <span class="type">int</span> maxSize = <span class="number">10</span>;              <span class="comment">//指向结点的指针的向量/栈的初始化大小    大小&gt;=结点数</span></span><br><span class="line">    <span class="function">vector&lt;<span class="keyword">struct</span> BTNode *&gt; <span class="title">stack1</span><span class="params">(maxSize)</span></span>; <span class="comment">//指向结点的指针的向量/栈1 反前序遍历使用</span></span><br><span class="line">    <span class="function">vector&lt;<span class="keyword">struct</span> BTNode *&gt; <span class="title">stack2</span><span class="params">(maxSize)</span></span>; <span class="comment">//指向结点的指针的向量/栈2 逆逆后序遍历使用</span></span><br><span class="line">    <span class="type">int</span> top1 = <span class="number">-1</span>;                           <span class="comment">//指向栈1顶结点的指针1</span></span><br><span class="line">    <span class="type">int</span> top2 = <span class="number">-1</span>;                           <span class="comment">//指向栈2顶结点的指针2</span></span><br><span class="line"></span><br><span class="line">    <span class="keyword">struct</span> <span class="title class_">BTNode</span> *visitNode = <span class="literal">nullptr</span>; <span class="comment">//指向访问结点的指针</span></span><br><span class="line"></span><br><span class="line">    <span class="keyword">if</span> (root == <span class="literal">nullptr</span>) <span class="comment">//空树</span></span><br><span class="line">    &#123;</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="keyword">else</span> <span class="comment">//非空树</span></span><br><span class="line">    &#123;</span><br><span class="line">        <span class="comment">//反前序遍历过程</span></span><br><span class="line">        ++top1;</span><br><span class="line">        stack1[top1] = root; <span class="comment">//根结点入栈1</span></span><br><span class="line"></span><br><span class="line">        <span class="keyword">while</span> (top1 != <span class="number">-1</span>) <span class="comment">//栈1不为空时，循环</span></span><br><span class="line">        &#123;</span><br><span class="line">            visitNode = stack1[top1]; <span class="comment">//指向访问结点的指针指向栈1顶结点</span></span><br><span class="line">            --top1;                   <span class="comment">//栈1顶结点出栈1</span></span><br><span class="line"></span><br><span class="line">            ++top2;</span><br><span class="line">            stack2[top2] = visitNode; <span class="comment">//访问结点入栈2</span></span><br><span class="line"></span><br><span class="line">            <span class="comment">//反前序遍历与前序遍历操作相反</span></span><br><span class="line">            <span class="keyword">if</span> (visitNode-&gt;lChild != <span class="literal">nullptr</span>) <span class="comment">//左孩子结点不为空</span></span><br><span class="line">            &#123;</span><br><span class="line">                ++top1;</span><br><span class="line">                stack1[top1] = visitNode-&gt;lChild; <span class="comment">//左孩子结点入栈1</span></span><br><span class="line">            &#125;</span><br><span class="line"></span><br><span class="line">            <span class="keyword">if</span> (visitNode-&gt;rChild != <span class="literal">nullptr</span>) <span class="comment">//右孩子结点不为空</span></span><br><span class="line">            &#123;</span><br><span class="line">                ++top1;</span><br><span class="line">                stack1[top1] = visitNode-&gt;rChild; <span class="comment">//右孩子结点入栈1</span></span><br><span class="line">            &#125;</span><br><span class="line">        &#125;</span><br><span class="line"></span><br><span class="line">        <span class="comment">//逆逆后序遍历过程</span></span><br><span class="line">        <span class="keyword">while</span> (top2 != <span class="number">-1</span>) <span class="comment">//栈2不为空时，循环</span></span><br><span class="line">        &#123;</span><br><span class="line">            visitNode = stack2[top2]; <span class="comment">//指向访问结点的指针指向栈2顶结点</span></span><br><span class="line">            --top2;                   <span class="comment">//栈2顶结点出栈2</span></span><br><span class="line"></span><br><span class="line">            cout &lt;&lt; visitNode-&gt;data; <span class="comment">//访问结点的数据</span></span><br><span class="line">        &#125;</span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line">    <span class="keyword">return</span>;</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="comment">//层序遍历  迭代法</span></span><br><span class="line"><span class="comment">//时间复杂度：O(n)。n为数据规模</span></span><br><span class="line"><span class="comment">//空间复杂度：O(n)。n为数据规模</span></span><br><span class="line"><span class="comment">//空间复杂度：指向结点的指针的向量/循环队列的大小</span></span><br><span class="line"><span class="function"><span class="type">void</span> <span class="title">levelOrderTraverse2</span><span class="params">(<span class="keyword">struct</span> BTNode *root)</span> <span class="comment">//参数：根结点指针</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="keyword">constexpr</span> <span class="type">int</span> maxSize = <span class="number">10</span>;             <span class="comment">//指向结点的指针的向量/循环队列的初始化大小    大小尽量&gt;=结点数</span></span><br><span class="line">    <span class="function">vector&lt;<span class="keyword">struct</span> BTNode *&gt; <span class="title">queue</span><span class="params">(maxSize)</span></span>; <span class="comment">//指向结点的指针的向量/循环队列</span></span><br><span class="line">    <span class="type">int</span> front = <span class="number">0</span>;                          <span class="comment">//指向队列头结点的指针</span></span><br><span class="line">    <span class="type">int</span> rear = <span class="number">0</span>;                           <span class="comment">//指向队列尾结点的下一结点的指针</span></span><br><span class="line"></span><br><span class="line">    <span class="keyword">struct</span> <span class="title class_">BTNode</span> *visitNode = <span class="literal">nullptr</span>; <span class="comment">//指向访问结点的指针</span></span><br><span class="line"></span><br><span class="line">    <span class="keyword">if</span> (root == <span class="literal">nullptr</span>) <span class="comment">//空树</span></span><br><span class="line">    &#123;</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="keyword">else</span> <span class="comment">//非空树</span></span><br><span class="line">    &#123;</span><br><span class="line">        queue[rear] = root; <span class="comment">//根结点入队</span></span><br><span class="line">        rear = (rear + <span class="number">1</span>) % maxSize;</span><br><span class="line"></span><br><span class="line">        <span class="keyword">while</span> (front != rear) <span class="comment">//队列不为空，循环</span></span><br><span class="line">        &#123;</span><br><span class="line">            visitNode = queue[front];      <span class="comment">//指向访问结点的指针指向队列头结点</span></span><br><span class="line">            front = (front + <span class="number">1</span>) % maxSize; <span class="comment">//队列头结点出队</span></span><br><span class="line"></span><br><span class="line">            cout &lt;&lt; visitNode-&gt;data; <span class="comment">//访问结点的数据</span></span><br><span class="line"></span><br><span class="line">            <span class="keyword">if</span> (visitNode-&gt;lChild != <span class="literal">nullptr</span>) <span class="comment">//左子树不为空</span></span><br><span class="line">            &#123;</span><br><span class="line">                queue[rear] = visitNode-&gt;lChild; <span class="comment">//左子树的根结点入队</span></span><br><span class="line">                rear = (rear + <span class="number">1</span>) % maxSize;</span><br><span class="line">            &#125;</span><br><span class="line"></span><br><span class="line">            <span class="keyword">if</span> (visitNode-&gt;rChild != <span class="literal">nullptr</span>) <span class="comment">//右子树不为空</span></span><br><span class="line">            &#123;</span><br><span class="line">                queue[rear] = visitNode-&gt;rChild; <span class="comment">//右子树的根结点入队</span></span><br><span class="line">                rear = (rear + <span class="number">1</span>) % maxSize;</span><br><span class="line">            &#125;</span><br><span class="line">        &#125;</span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line">    <span class="keyword">return</span>;</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="comment">//获取深度/高度</span></span><br><span class="line"><span class="comment">//本质：后序遍历</span></span><br><span class="line"><span class="comment">//时间复杂度：O(n)。n为数据规模</span></span><br><span class="line"><span class="comment">//空间复杂度：O(n)。n为数据规模</span></span><br><span class="line"><span class="comment">//空间复杂度：递归调用栈的规模/树的深度/高度：</span></span><br><span class="line"><span class="comment">//最好：[log以2为底的n]下取整+1或[log以2为底的(n+1)]上取整。n为数据规模</span></span><br><span class="line"><span class="comment">//最坏：n。n为数据规模</span></span><br><span class="line"><span class="function"><span class="type">int</span> <span class="title">getDepth</span><span class="params">(<span class="keyword">struct</span> BTNode *root)</span> <span class="comment">//参数：根结点指针</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="type">int</span> lDepth = <span class="number">0</span>; <span class="comment">//左子树的深度</span></span><br><span class="line">    <span class="type">int</span> rDepth = <span class="number">0</span>; <span class="comment">//右子树的深度</span></span><br><span class="line">    <span class="type">int</span> depth = <span class="number">0</span>;  <span class="comment">//深度=左子树深度和右子树深度间的最大值+1</span></span><br><span class="line"></span><br><span class="line">    <span class="keyword">if</span> (root == <span class="literal">nullptr</span>) <span class="comment">//空树</span></span><br><span class="line">    &#123;</span><br><span class="line">        depth = <span class="number">0</span>;</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="keyword">else</span> <span class="comment">//非空树</span></span><br><span class="line">    &#123;</span><br><span class="line">        lDepth = <span class="built_in">getDepth</span>(root-&gt;lChild); <span class="comment">//后序遍历获取左子树的深度</span></span><br><span class="line">        rDepth = <span class="built_in">getDepth</span>(root-&gt;rChild); <span class="comment">//后序遍历获取右子树的深度</span></span><br><span class="line">        <span class="keyword">if</span> (lDepth &lt; rDepth)</span><br><span class="line">        &#123;</span><br><span class="line">            depth = rDepth + <span class="number">1</span>;</span><br><span class="line">        &#125;</span><br><span class="line">        <span class="keyword">else</span></span><br><span class="line">        &#123;</span><br><span class="line">            depth = lDepth + <span class="number">1</span>;</span><br><span class="line">        &#125;</span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line">    <span class="keyword">return</span> depth;</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="comment">//主函数————————————————————</span></span><br><span class="line"><span class="function"><span class="type">int</span> <span class="title">main</span><span class="params">()</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="built_in">use_example</span>(); <span class="comment">//使用示例</span></span><br><span class="line"></span><br><span class="line">    <span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>
<hr>
<h1 id="tBTree-cpp（线索二叉树）"><a href="#tBTree-cpp（线索二叉树）" class="headerlink" title="tBTree.cpp（线索二叉树）"></a>tBTree.cpp（线索二叉树）</h1><figure class="highlight cpp"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br><span class="line">74</span><br><span class="line">75</span><br><span class="line">76</span><br><span class="line">77</span><br><span class="line">78</span><br><span class="line">79</span><br><span class="line">80</span><br><span class="line">81</span><br><span class="line">82</span><br><span class="line">83</span><br><span class="line">84</span><br><span class="line">85</span><br><span class="line">86</span><br><span class="line">87</span><br><span 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class="line">148</span><br><span class="line">149</span><br><span class="line">150</span><br><span class="line">151</span><br><span class="line">152</span><br><span class="line">153</span><br><span class="line">154</span><br><span class="line">155</span><br><span class="line">156</span><br><span class="line">157</span><br><span class="line">158</span><br><span class="line">159</span><br><span class="line">160</span><br><span class="line">161</span><br><span class="line">162</span><br><span class="line">163</span><br><span class="line">164</span><br><span class="line">165</span><br><span class="line">166</span><br><span class="line">167</span><br><span class="line">168</span><br><span class="line">169</span><br><span class="line">170</span><br><span class="line">171</span><br><span class="line">172</span><br><span class="line">173</span><br><span class="line">174</span><br><span class="line">175</span><br><span class="line">176</span><br><span class="line">177</span><br><span class="line">178</span><br><span class="line">179</span><br><span class="line">180</span><br><span class="line">181</span><br><span class="line">182</span><br><span class="line">183</span><br><span class="line">184</span><br><span class="line">185</span><br><span class="line">186</span><br><span class="line">187</span><br><span class="line">188</span><br><span class="line">189</span><br><span class="line">190</span><br><span class="line">191</span><br><span class="line">192</span><br><span class="line">193</span><br><span class="line">194</span><br><span class="line">195</span><br><span class="line">196</span><br><span class="line">197</span><br><span class="line">198</span><br><span class="line">199</span><br><span class="line">200</span><br><span class="line">201</span><br><span class="line">202</span><br><span class="line">203</span><br><span class="line">204</span><br><span class="line">205</span><br><span class="line">206</span><br><span class="line">207</span><br><span class="line">208</span><br><span class="line">209</span><br><span class="line">210</span><br><span class="line">211</span><br><span class="line">212</span><br><span class="line">213</span><br><span class="line">214</span><br><span class="line">215</span><br><span class="line">216</span><br><span class="line">217</span><br><span class="line">218</span><br><span class="line">219</span><br><span class="line">220</span><br><span class="line">221</span><br><span class="line">222</span><br><span class="line">223</span><br><span class="line">224</span><br><span class="line">225</span><br><span class="line">226</span><br><span class="line">227</span><br><span class="line">228</span><br><span class="line">229</span><br><span class="line">230</span><br><span class="line">231</span><br><span class="line">232</span><br><span class="line">233</span><br><span class="line">234</span><br><span class="line">235</span><br><span class="line">236</span><br><span class="line">237</span><br><span class="line">238</span><br><span class="line">239</span><br><span class="line">240</span><br><span class="line">241</span><br><span class="line">242</span><br><span class="line">243</span><br><span class="line">244</span><br><span class="line">245</span><br><span class="line">246</span><br><span class="line">247</span><br><span class="line">248</span><br><span class="line">249</span><br><span class="line">250</span><br><span class="line">251</span><br><span class="line">252</span><br><span class="line">253</span><br><span class="line">254</span><br><span class="line">255</span><br><span class="line">256</span><br><span class="line">257</span><br><span class="line">258</span><br><span class="line">259</span><br><span class="line">260</span><br><span class="line">261</span><br><span class="line">262</span><br><span class="line">263</span><br><span class="line">264</span><br><span class="line">265</span><br><span class="line">266</span><br><span class="line">267</span><br><span class="line">268</span><br><span class="line">269</span><br><span class="line">270</span><br><span class="line">271</span><br><span class="line">272</span><br><span class="line">273</span><br><span class="line">274</span><br><span class="line">275</span><br><span class="line">276</span><br><span class="line">277</span><br><span class="line">278</span><br><span class="line">279</span><br><span class="line">280</span><br><span class="line">281</span><br><span class="line">282</span><br><span class="line">283</span><br><span class="line">284</span><br><span class="line">285</span><br><span class="line">286</span><br><span class="line">287</span><br><span class="line">288</span><br><span class="line">289</span><br><span class="line">290</span><br><span class="line">291</span><br><span class="line">292</span><br><span class="line">293</span><br><span class="line">294</span><br><span class="line">295</span><br><span class="line">296</span><br><span class="line">297</span><br><span class="line">298</span><br><span class="line">299</span><br><span class="line">300</span><br><span class="line">301</span><br><span class="line">302</span><br><span class="line">303</span><br><span class="line">304</span><br><span class="line">305</span><br><span class="line">306</span><br><span class="line">307</span><br><span class="line">308</span><br><span class="line">309</span><br><span class="line">310</span><br><span class="line">311</span><br><span class="line">312</span><br><span class="line">313</span><br><span class="line">314</span><br><span class="line">315</span><br><span class="line">316</span><br><span class="line">317</span><br><span class="line">318</span><br><span class="line">319</span><br><span class="line">320</span><br><span class="line">321</span><br><span class="line">322</span><br><span class="line">323</span><br><span class="line">324</span><br><span class="line">325</span><br><span class="line">326</span><br><span class="line">327</span><br><span class="line">328</span><br><span class="line">329</span><br><span class="line">330</span><br><span class="line">331</span><br><span class="line">332</span><br><span class="line">333</span><br><span class="line">334</span><br><span class="line">335</span><br><span class="line">336</span><br><span class="line">337</span><br><span class="line">338</span><br><span class="line">339</span><br><span class="line">340</span><br><span class="line">341</span><br><span class="line">342</span><br><span class="line">343</span><br><span class="line">344</span><br><span class="line">345</span><br><span class="line">346</span><br><span class="line">347</span><br><span class="line">348</span><br><span class="line">349</span><br><span class="line">350</span><br><span class="line">351</span><br><span class="line">352</span><br><span class="line">353</span><br><span class="line">354</span><br><span class="line">355</span><br></pre></td><td class="code"><pre><span class="line"><span class="comment">//头文件————————————————————</span></span><br><span class="line"><span class="meta">#<span class="keyword">include</span> <span class="string">&lt;iostream&gt;</span></span></span><br><span class="line"></span><br><span class="line"><span class="comment">//命名空间</span></span><br><span class="line"><span class="keyword">using</span> <span class="keyword">namespace</span> std;</span><br><span class="line"></span><br><span class="line"><span class="comment">//自定义数据类型————————————————————</span></span><br><span class="line"><span class="keyword">typedef</span> <span class="type">int</span> ELEM_TYPE; <span class="comment">//数据的（数据）类型 依据数据的实际类型定义</span></span><br><span class="line"></span><br><span class="line"><span class="comment">//结构体</span></span><br><span class="line"><span class="comment">//结点</span></span><br><span class="line"><span class="keyword">typedef</span> <span class="keyword">struct</span> <span class="title class_">TBTNode</span></span><br><span class="line">&#123;</span><br><span class="line">    ELEM_TYPE data; <span class="comment">//数据</span></span><br><span class="line"></span><br><span class="line">    <span class="type">int</span> lTag;               <span class="comment">//左线索标志    可用bool数据类型</span></span><br><span class="line">    <span class="type">int</span> rTag;               <span class="comment">//右线索标志</span></span><br><span class="line">    <span class="keyword">struct</span> <span class="title class_">TBTNode</span> *lChild; <span class="comment">//指向左孩子结点的指针</span></span><br><span class="line">    <span class="keyword">struct</span> <span class="title class_">TBTNode</span> *rChild; <span class="comment">//指向右孩子结点的指针</span></span><br><span class="line">    <span class="comment">//约定：</span></span><br><span class="line">    <span class="comment">//当lTag=0时，lChild为指针，指向左孩子结点；当lTag=1时，lChild为线索，指向直接前驱结点</span></span><br><span class="line">    <span class="comment">//当rTag=0时，rChild为指针，指向右孩子结点；当rTag=1时，rChild为线索，指向直接后继结点</span></span><br><span class="line">&#125; TBTNode;</span><br><span class="line"><span class="comment">// struct TBTNode：结构体数据类型的名称</span></span><br><span class="line"><span class="comment">//如创建一个结点的语句：struct TBTNode node;</span></span><br><span class="line"><span class="comment">//若结构体数据类型内，存在指向该结构体的指针数据类型，则必须完整命名结构体数据类型的名称</span></span><br><span class="line"><span class="comment">//如结点结构体TBTNode内，存在指向该结构体的指针数据类型lChild和rChild，则命名语句为：typedef struct TBTNode而不是typedef struct</span></span><br><span class="line"><span class="comment">// &#125;TBTNode;：typedef给结构体数据类型起的别名，可简化语句</span></span><br><span class="line"><span class="comment">//如创建一个结点的语句：TBTNode node;</span></span><br><span class="line"><span class="comment">//结构体数据类型的名称和别名尽量一致，以方便记忆、使用</span></span><br><span class="line"></span><br><span class="line"><span class="comment">//函数声明————————————————————</span></span><br><span class="line"><span class="function"><span class="type">void</span> <span class="title">use_example</span><span class="params">()</span></span>;                       <span class="comment">//使用示例</span></span><br><span class="line"><span class="function"><span class="type">void</span> <span class="title">createTBTree</span><span class="params">(<span class="keyword">struct</span> TBTNode *&amp;root)</span></span>; <span class="comment">//创建 前序法</span></span><br><span class="line"></span><br><span class="line"><span class="function"><span class="type">void</span> <span class="title">inThread</span><span class="params">(<span class="keyword">struct</span> TBTNode *cur, <span class="keyword">struct</span> TBTNode *&amp;pre)</span></span>; <span class="comment">//线索化    中序遍历法</span></span><br><span class="line"><span class="function"><span class="type">void</span> <span class="title">createInThread</span><span class="params">(<span class="keyword">struct</span> TBTNode *root)</span></span>;                <span class="comment">//线索化创建    中序遍历法</span></span><br><span class="line"><span class="keyword">struct</span> <span class="title class_">TBTNode</span> *<span class="built_in">inFirst</span>(<span class="keyword">struct</span> TBTNode *cur);             <span class="comment">//获取第一个结点    中序遍历法</span></span><br><span class="line"><span class="keyword">struct</span> <span class="title class_">TBTNode</span> *<span class="built_in">inNext</span>(<span class="keyword">struct</span> TBTNode *cur);              <span class="comment">//获取后继结点    中序遍历法</span></span><br><span class="line"><span class="function"><span class="type">void</span> <span class="title">inOrderTraverse</span><span class="params">(<span class="keyword">struct</span> TBTNode *root)</span></span>;               <span class="comment">//中序遍历</span></span><br><span class="line"></span><br><span class="line"><span class="function"><span class="type">void</span> <span class="title">preThread</span><span class="params">(<span class="keyword">struct</span> TBTNode *cur, <span class="keyword">struct</span> TBTNode *&amp;pre)</span></span>; <span class="comment">//线索化    前序遍历法</span></span><br><span class="line"><span class="function"><span class="type">void</span> <span class="title">preOrderTraverse</span><span class="params">(<span class="keyword">struct</span> TBTNode *root)</span></span>;               <span class="comment">//前序遍历</span></span><br><span class="line"></span><br><span class="line"><span class="function"><span class="type">void</span> <span class="title">postThread</span><span class="params">(<span class="keyword">struct</span> TBTNode *cur, <span class="keyword">struct</span> TBTNode *&amp;pre)</span></span>; <span class="comment">//线索化    后序遍历法</span></span><br><span class="line"><span class="comment">//后序遍历：复杂，需要知道结点双亲-&gt;使用带标志域的三叉链表</span></span><br><span class="line"></span><br><span class="line"><span class="comment">//函数定义————————————————————</span></span><br><span class="line"><span class="comment">//使用示例</span></span><br><span class="line"><span class="function"><span class="type">void</span> <span class="title">use_example</span><span class="params">()</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="keyword">struct</span> <span class="title class_">TBTNode</span> *root; <span class="comment">//根结点指针</span></span><br><span class="line"></span><br><span class="line">    <span class="built_in">createTBTree</span>(root); <span class="comment">//创建 前序法</span></span><br><span class="line">    <span class="comment">//构造三层二叉树：</span></span><br><span class="line">    <span class="comment">//第一层：1</span></span><br><span class="line">    <span class="comment">//第二层：2，3</span></span><br><span class="line">    <span class="comment">//第三层：-1，-1，-1，4</span></span><br><span class="line">    <span class="comment">//输入数据：1，2，-1，-1，3，-1，4，-1，-1</span></span><br><span class="line"></span><br><span class="line">    <span class="built_in">createInThread</span>(root); <span class="comment">//线索化创建    中序遍历法</span></span><br><span class="line"></span><br><span class="line">    <span class="built_in">inOrderTraverse</span>(root); <span class="comment">//中序遍历  输出：2，1，3，4</span></span><br><span class="line">    cout &lt;&lt; endl;</span><br><span class="line"></span><br><span class="line">    <span class="keyword">struct</span> <span class="title class_">TBTNode</span> *root1; <span class="comment">//根结点指针1</span></span><br><span class="line"></span><br><span class="line">    <span class="built_in">createTBTree</span>(root1); <span class="comment">//创建 前序法</span></span><br><span class="line">    <span class="comment">//构造三层二叉树：</span></span><br><span class="line">    <span class="comment">//第一层：1</span></span><br><span class="line">    <span class="comment">//第二层：2，3</span></span><br><span class="line">    <span class="comment">//第三层：-1，-1，-1，4</span></span><br><span class="line">    <span class="comment">//输入数据：1，2，-1，-1，3，-1，4，-1，-1</span></span><br><span class="line"></span><br><span class="line">    <span class="keyword">struct</span> <span class="title class_">TBTNode</span> *pre = <span class="literal">nullptr</span>; <span class="comment">//前驱结点指针  指向空</span></span><br><span class="line">    <span class="built_in">preThread</span>(root1, pre);         <span class="comment">//线索化    前序遍历法</span></span><br><span class="line"></span><br><span class="line">    <span class="built_in">preOrderTraverse</span>(root1); <span class="comment">//前序遍历  输入：1，2，3，4</span></span><br><span class="line">    cout &lt;&lt; endl;</span><br><span class="line"></span><br><span class="line">    <span class="keyword">struct</span> <span class="title class_">TBTNode</span> *root2; <span class="comment">//根结点指针2</span></span><br><span class="line"></span><br><span class="line">    <span class="built_in">createTBTree</span>(root2); <span class="comment">//创建 前序法</span></span><br><span class="line">    <span class="comment">//构造三层二叉树：</span></span><br><span class="line">    <span class="comment">//第一层：1</span></span><br><span class="line">    <span class="comment">//第二层：2，3</span></span><br><span class="line">    <span class="comment">//第三层：-1，-1，-1，4</span></span><br><span class="line">    <span class="comment">//输入数据：1，2，-1，-1，3，-1，4，-1，-1</span></span><br><span class="line"></span><br><span class="line">    <span class="keyword">struct</span> <span class="title class_">TBTNode</span> *pre1 = <span class="literal">nullptr</span>; <span class="comment">//前驱结点指针1  指向空</span></span><br><span class="line">    <span class="built_in">postThread</span>(root2, pre1);        <span class="comment">//线索化    后序遍历法</span></span><br><span class="line"></span><br><span class="line">    <span class="keyword">return</span>;</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="comment">//创建 前序法</span></span><br><span class="line"><span class="comment">//时间复杂度：O(n)。n为数据规模</span></span><br><span class="line"><span class="comment">//空间复杂度：O(n)。n为数据规模</span></span><br><span class="line"><span class="comment">//空间复杂度：递归调用栈的规模/树的深度/高度：</span></span><br><span class="line"><span class="comment">//最好：[log以2为底的n]下取整+1或[log以2为底的(n+1)]上取整。n为数据规模</span></span><br><span class="line"><span class="comment">//最坏：n。n为数据规模</span></span><br><span class="line"><span class="comment">//注意：参数使用*&amp;，指向指针的引用数据类型</span></span><br><span class="line"><span class="comment">//若只使用*：指针数据类型，在函数体中并未进行解引用调用修改*root的值，使用malloc()修改的是root的值</span></span><br><span class="line"><span class="comment">//所以是值拷贝，函数返回后数据丢失。地址拷贝需要引用数据类型的配合</span></span><br><span class="line"><span class="function"><span class="type">void</span> <span class="title">createTBTree</span><span class="params">(<span class="keyword">struct</span> TBTNode *&amp;root)</span> <span class="comment">//参数：根结点指针</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    ELEM_TYPE value = <span class="number">0</span>; <span class="comment">//值</span></span><br><span class="line">    cin &gt;&gt; value;        <span class="comment">//获取输入值</span></span><br><span class="line"></span><br><span class="line">    <span class="keyword">if</span> (value == <span class="number">-1</span>) <span class="comment">//空树  约定值为-1时无结点</span></span><br><span class="line">    &#123;</span><br><span class="line">        root = <span class="literal">nullptr</span>; <span class="comment">//根结点指针指向空</span></span><br><span class="line">    &#125;</span><br><span class="line">    <span class="keyword">else</span> <span class="comment">//非空树</span></span><br><span class="line">    &#123;</span><br><span class="line">        root = (<span class="keyword">struct</span> TBTNode *)<span class="built_in">malloc</span>(<span class="built_in">sizeof</span>(<span class="keyword">struct</span> TBTNode)); <span class="comment">//创建根结点，根结点指针指向根结点</span></span><br><span class="line"></span><br><span class="line">        root-&gt;data = value;         <span class="comment">//初始化根结点的数据域</span></span><br><span class="line">        root-&gt;lTag = <span class="number">0</span>;             <span class="comment">//初始化根结点的左线索标志</span></span><br><span class="line">        root-&gt;rTag = <span class="number">0</span>;             <span class="comment">//初始化右结点的左线索标志</span></span><br><span class="line">        <span class="built_in">createTBTree</span>(root-&gt;lChild); <span class="comment">//递归构造左子树</span></span><br><span class="line">        <span class="built_in">createTBTree</span>(root-&gt;rChild); <span class="comment">//递归构造右子树</span></span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line">    <span class="keyword">return</span>;</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="comment">//线索化    中序遍历法</span></span><br><span class="line"><span class="comment">//时间复杂度：O(n)。n为数据规模</span></span><br><span class="line"><span class="comment">//空间复杂度：O(n)。n为数据规模</span></span><br><span class="line"><span class="comment">//空间复杂度：递归调用栈的规模/树的深度/高度：</span></span><br><span class="line"><span class="comment">//最好：[log以2为底的n]下取整+1或[log以2为底的(n+1)]上取整。n为数据规模</span></span><br><span class="line"><span class="comment">//最坏：n。n为数据规模</span></span><br><span class="line"><span class="function"><span class="type">void</span> <span class="title">inThread</span><span class="params">(<span class="keyword">struct</span> TBTNode *cur, <span class="keyword">struct</span> TBTNode *&amp;pre)</span> <span class="comment">//参数：当前结点指针，前驱结点指针</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="keyword">if</span> (cur == <span class="literal">nullptr</span>) <span class="comment">//空树</span></span><br><span class="line">    &#123;</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="keyword">else</span> <span class="comment">//非空树</span></span><br><span class="line">    &#123;</span><br><span class="line">        <span class="built_in">inThread</span>(cur-&gt;lChild, pre); <span class="comment">//当前结点的左子树线索化</span></span><br><span class="line"></span><br><span class="line">        <span class="comment">//线索化</span></span><br><span class="line">        <span class="keyword">if</span> (cur-&gt;lChild == <span class="literal">nullptr</span>) <span class="comment">//当前结点的左孩子结点为空   建立当前结点的前驱线索</span></span><br><span class="line">        &#123;</span><br><span class="line">            cur-&gt;lTag = <span class="number">1</span>;     <span class="comment">//当前结点的左线索标志置1</span></span><br><span class="line">            cur-&gt;lChild = pre; <span class="comment">//当前结点的左孩子结点指针指向前驱结点</span></span><br><span class="line">        &#125;</span><br><span class="line">        <span class="keyword">if</span> (pre != <span class="literal">nullptr</span> &amp;&amp; pre-&gt;rChild == <span class="literal">nullptr</span>) <span class="comment">//前驱结点不为空且前驱结点的右孩子结点为空   建立前驱结点的后继线索</span></span><br><span class="line">        &#123;</span><br><span class="line">            pre-&gt;rTag = <span class="number">1</span>;     <span class="comment">//前驱结点的右线索标志置1</span></span><br><span class="line">            pre-&gt;rChild = cur; <span class="comment">//前驱结点的右孩子结点指针指向当前结点</span></span><br><span class="line">        &#125;</span><br><span class="line"></span><br><span class="line">        pre = cur; <span class="comment">//更新结点指针：前驱结点指针指向当前结点</span></span><br><span class="line"></span><br><span class="line">        <span class="built_in">inThread</span>(cur-&gt;rChild, pre); <span class="comment">//当前结点的右子树线索化</span></span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line">    <span class="keyword">return</span>;</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="comment">//线索化创建    中序遍历法</span></span><br><span class="line"><span class="comment">//时间复杂度：O(n)。n为数据规模</span></span><br><span class="line"><span class="comment">//空间复杂度：O(n)。n为数据规模</span></span><br><span class="line"><span class="comment">//空间复杂度：递归调用栈的规模/树的深度/高度：</span></span><br><span class="line"><span class="comment">//最好：[log以2为底的n]下取整+1或[log以2为底的(n+1)]上取整。n为数据规模</span></span><br><span class="line"><span class="comment">//最坏：n。n为数据规模</span></span><br><span class="line"><span class="function"><span class="type">void</span> <span class="title">createInThread</span><span class="params">(<span class="keyword">struct</span> TBTNode *root)</span> <span class="comment">//参数：根结点指针</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="keyword">struct</span> <span class="title class_">TBTNode</span> *pre = <span class="literal">nullptr</span>; <span class="comment">//前驱结点指针  指向空</span></span><br><span class="line"></span><br><span class="line">    <span class="keyword">if</span> (root == <span class="literal">nullptr</span>) <span class="comment">//空树</span></span><br><span class="line">    &#123;</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="keyword">else</span> <span class="comment">//非空树</span></span><br><span class="line">    &#123;</span><br><span class="line">        <span class="built_in">inThread</span>(root, pre); <span class="comment">//线索化 中序遍历法</span></span><br><span class="line"></span><br><span class="line">        <span class="comment">//已更新结点指针：前驱结点指针指向最后一个结点</span></span><br><span class="line">        <span class="comment">//处理最后一个结点：必为叶子结点，右孩子结点为空</span></span><br><span class="line">        pre-&gt;rTag = <span class="number">1</span>;</span><br><span class="line">        pre-&gt;rChild = <span class="literal">nullptr</span>;</span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line">    <span class="keyword">return</span>;</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="comment">//获取第一个结点    中序遍历法</span></span><br><span class="line"><span class="comment">//时间复杂度：O(n)。n为数据规模</span></span><br><span class="line"><span class="comment">//时间复杂度：树的深度/高度：</span></span><br><span class="line"><span class="comment">//最好：[log以2为底的n]下取整+1或[log以2为底的(n+1)]上取整。n为数据规模</span></span><br><span class="line"><span class="comment">//最坏：n。n为数据规模</span></span><br><span class="line"><span class="comment">//空间复杂度：O(1)。未使用辅助空间</span></span><br><span class="line"><span class="keyword">struct</span> <span class="title class_">TBTNode</span> *<span class="built_in">inFirst</span>(<span class="keyword">struct</span> TBTNode *cur) <span class="comment">//参数：当前结点指针    返回值：以当前结点为根的树，中序遍历时的第一个结点</span></span><br><span class="line">&#123;</span><br><span class="line">    <span class="comment">//以当前结点为根的树，中序遍历的第一个结点是最左下结点</span></span><br><span class="line">    <span class="keyword">while</span> (cur-&gt;lTag == <span class="number">0</span>) <span class="comment">//当前结点的左线索标志为0</span></span><br><span class="line">    &#123;</span><br><span class="line">        cur = cur-&gt;lChild; <span class="comment">//更新当前结点指针指向当前结点的左孩子结点</span></span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line">    <span class="keyword">return</span> cur;</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="comment">//获取后继结点    中序遍历法</span></span><br><span class="line"><span class="comment">//时间复杂度：O(n)。n为数据规模</span></span><br><span class="line"><span class="comment">//时间复杂度：树的深度/高度或直接获取</span></span><br><span class="line"><span class="comment">//最好：[log以2为底的n]下取整+1或[log以2为底的(n+1)]上取整。n为数据规模。或O(1)</span></span><br><span class="line"><span class="comment">//最坏：n。n为数据规模</span></span><br><span class="line"><span class="comment">//空间复杂度：O(1)。未使用辅助空间</span></span><br><span class="line"><span class="keyword">struct</span> <span class="title class_">TBTNode</span> *<span class="built_in">inNext</span>(<span class="keyword">struct</span> TBTNode *cur) <span class="comment">//参数：当前结点指针    返回值：中序遍历时，当前结点的后继结点</span></span><br><span class="line">&#123;</span><br><span class="line">    <span class="comment">//若当前结点的右线索标志为0，存在以当前结点的右孩子结点为根的右子树，获取第一个结点</span></span><br><span class="line">    <span class="comment">//若当前结点的右线索标志为1，右孩子结点指针指向直接后继结点，直接获取</span></span><br><span class="line">    <span class="keyword">if</span> (cur-&gt;rTag == <span class="number">0</span>)</span><br><span class="line">    &#123;</span><br><span class="line">        <span class="keyword">return</span> <span class="built_in">inFirst</span>(cur-&gt;rChild);</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="keyword">else</span></span><br><span class="line">    &#123;</span><br><span class="line">        <span class="keyword">return</span> cur-&gt;rChild;</span><br><span class="line">    &#125;</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="comment">//中序遍历</span></span><br><span class="line"><span class="comment">//时间复杂度：O(n)。n为数据规模</span></span><br><span class="line"><span class="comment">//空间复杂度：O(1)。未使用辅助空间</span></span><br><span class="line"><span class="function"><span class="type">void</span> <span class="title">inOrderTraverse</span><span class="params">(<span class="keyword">struct</span> TBTNode *root)</span> <span class="comment">//参数：根结点指针</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="keyword">for</span> (<span class="keyword">struct</span> TBTNode *cur = <span class="built_in">inFirst</span>(root); cur != <span class="literal">nullptr</span>; cur = <span class="built_in">inNext</span>(cur))</span><br><span class="line">    &#123;</span><br><span class="line">        cout &lt;&lt; cur-&gt;data; <span class="comment">//访问结点的数据</span></span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line">    <span class="keyword">return</span>;</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="comment">//线索化    前序遍历法</span></span><br><span class="line"><span class="comment">//时间复杂度：O(n)。n为数据规模</span></span><br><span class="line"><span class="comment">//空间复杂度：O(n)。n为数据规模</span></span><br><span class="line"><span class="comment">//空间复杂度：递归调用栈的规模/树的深度/高度：</span></span><br><span class="line"><span class="comment">//最好：[log以2为底的n]下取整+1或[log以2为底的(n+1)]上取整。n为数据规模</span></span><br><span class="line"><span class="comment">//最坏：n。n为数据规模</span></span><br><span class="line"><span class="function"><span class="type">void</span> <span class="title">preThread</span><span class="params">(<span class="keyword">struct</span> TBTNode *cur, <span class="keyword">struct</span> TBTNode *&amp;pre)</span> <span class="comment">//参数：当前结点指针，前驱结点指针</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="keyword">if</span> (cur == <span class="literal">nullptr</span>) <span class="comment">//空树</span></span><br><span class="line">    &#123;</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="keyword">else</span> <span class="comment">//非空树</span></span><br><span class="line">    &#123;</span><br><span class="line">        <span class="comment">//线索化</span></span><br><span class="line">        <span class="keyword">if</span> (cur-&gt;lChild == <span class="literal">nullptr</span>) <span class="comment">//当前结点的左孩子结点为空   建立当前结点的前驱线索</span></span><br><span class="line">        &#123;</span><br><span class="line">            cur-&gt;lTag = <span class="number">1</span>;     <span class="comment">//当前结点的左线索标志置1</span></span><br><span class="line">            cur-&gt;lChild = pre; <span class="comment">//当前结点的左孩子结点指针指向前驱结点</span></span><br><span class="line">        &#125;</span><br><span class="line">        <span class="keyword">if</span> (pre != <span class="literal">nullptr</span> &amp;&amp; pre-&gt;rChild == <span class="literal">nullptr</span>) <span class="comment">//前驱结点不为空且前驱结点的右孩子结点为空   建立前驱结点的后继线索</span></span><br><span class="line">        &#123;</span><br><span class="line">            pre-&gt;rTag = <span class="number">1</span>;     <span class="comment">//前驱结点的右线索标志置1</span></span><br><span class="line">            pre-&gt;rChild = cur; <span class="comment">//前驱结点的右孩子结点指针指向当前结点</span></span><br><span class="line">        &#125;</span><br><span class="line"></span><br><span class="line">        pre = cur; <span class="comment">//更新结点指针：前驱结点指针指向当前结点</span></span><br><span class="line"></span><br><span class="line">        <span class="comment">//注意：有限制条件</span></span><br><span class="line">        <span class="keyword">if</span> (cur-&gt;lTag == <span class="number">0</span>) <span class="comment">//当前结点的左线索标志为0，存在左孩子结点</span></span><br><span class="line">        &#123;</span><br><span class="line">            <span class="built_in">preThread</span>(cur-&gt;lChild, pre); <span class="comment">//当前结点的左子树线索化</span></span><br><span class="line">        &#125;</span><br><span class="line">        <span class="keyword">if</span> (cur-&gt;rTag == <span class="number">0</span>) <span class="comment">//当前结点的左线索标志为0，存在右孩子结点</span></span><br><span class="line">        &#123;</span><br><span class="line">            <span class="built_in">preThread</span>(cur-&gt;rChild, pre); <span class="comment">//当前结点的右子树线索化</span></span><br><span class="line">        &#125;</span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line">    <span class="keyword">return</span>;</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="comment">//前序遍历</span></span><br><span class="line"><span class="comment">//时间复杂度：O(n)。n为数据规模</span></span><br><span class="line"><span class="comment">//空间复杂度：O(1)。未使用辅助空间</span></span><br><span class="line"><span class="function"><span class="type">void</span> <span class="title">preOrderTraverse</span><span class="params">(<span class="keyword">struct</span> TBTNode *root)</span> <span class="comment">//参数：根结点指针</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="keyword">struct</span> <span class="title class_">TBTNode</span> *cur = <span class="literal">nullptr</span>; <span class="comment">//当前结点指针  指向空</span></span><br><span class="line"></span><br><span class="line">    <span class="keyword">if</span> (root == <span class="literal">nullptr</span>) <span class="comment">//空树</span></span><br><span class="line">    &#123;</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="keyword">else</span> <span class="comment">//非空树</span></span><br><span class="line">    &#123;</span><br><span class="line">        cur = root; <span class="comment">//当前结点指针指向根结点</span></span><br><span class="line"></span><br><span class="line">        <span class="keyword">while</span> (cur != <span class="literal">nullptr</span>) <span class="comment">//当前结点不为空时，说明未遍历完，循环</span></span><br><span class="line">        &#123;</span><br><span class="line">            <span class="keyword">while</span> (cur-&gt;lTag == <span class="number">0</span>) <span class="comment">//当前结点的左线索标志为0，左孩子结点不为空</span></span><br><span class="line">            &#123;</span><br><span class="line">                cout &lt;&lt; cur-&gt;data; <span class="comment">//访问当前结点的数据</span></span><br><span class="line">                cur = cur-&gt;lChild; <span class="comment">//当前结点指针指向当前结点的左孩子结点：中-&gt;左</span></span><br><span class="line">            &#125;</span><br><span class="line"></span><br><span class="line">            <span class="comment">//当前结点的左线索标志为1，左孩子结点为空</span></span><br><span class="line">            cout &lt;&lt; cur-&gt;data; <span class="comment">//访问当前结点的数据</span></span><br><span class="line">            cur = cur-&gt;rChild;</span><br><span class="line">            <span class="comment">//若当前结点的右线索标志为0，右孩子结点不为空，当前结点指针指向当前结点的右孩子结点：中-&gt;左（空）-&gt;右</span></span><br><span class="line">            <span class="comment">//若当前结点的右线索标志为1，右孩子结点为空，当前结点指针指向当前结点的直接后继结点</span></span><br><span class="line">        &#125;</span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line">    <span class="keyword">return</span>;</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="comment">//线索化    后序遍历法</span></span><br><span class="line"><span class="comment">//时间复杂度：O(n)。n为数据规模</span></span><br><span class="line"><span class="comment">//空间复杂度：O(n)。n为数据规模</span></span><br><span class="line"><span class="comment">//空间复杂度：递归调用栈的规模/树的深度/高度：</span></span><br><span class="line"><span class="comment">//最好：[log以2为底的n]下取整+1或[log以2为底的(n+1)]上取整。n为数据规模</span></span><br><span class="line"><span class="comment">//最坏：n。n为数据规模</span></span><br><span class="line"><span class="function"><span class="type">void</span> <span class="title">postThread</span><span class="params">(<span class="keyword">struct</span> TBTNode *cur, <span class="keyword">struct</span> TBTNode *&amp;pre)</span> <span class="comment">//参数：当前结点指针，前驱结点指针</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="keyword">if</span> (cur == <span class="literal">nullptr</span>) <span class="comment">//空树</span></span><br><span class="line">    &#123;</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="keyword">else</span> <span class="comment">//非空树</span></span><br><span class="line">    &#123;</span><br><span class="line"></span><br><span class="line">        <span class="built_in">postThread</span>(cur-&gt;lChild, pre); <span class="comment">//当前结点的左子树线索化</span></span><br><span class="line"></span><br><span class="line">        <span class="built_in">postThread</span>(cur-&gt;rChild, pre); <span class="comment">//当前结点的右子树线索化</span></span><br><span class="line"></span><br><span class="line">        <span class="comment">//线索化</span></span><br><span class="line">        <span class="keyword">if</span> (cur-&gt;lChild == <span class="literal">nullptr</span>) <span class="comment">//当前结点的左孩子结点为空   建立当前结点的前驱线索</span></span><br><span class="line">        &#123;</span><br><span class="line">            cur-&gt;lTag = <span class="number">1</span>;     <span class="comment">//当前结点的左线索标志置1</span></span><br><span class="line">            cur-&gt;lChild = pre; <span class="comment">//当前结点的左孩子结点指针指向前驱结点</span></span><br><span class="line">        &#125;</span><br><span class="line">        <span class="keyword">if</span> (pre != <span class="literal">nullptr</span> &amp;&amp; pre-&gt;rChild == <span class="literal">nullptr</span>) <span class="comment">//前驱结点不为空且前驱结点的右孩子结点为空   建立前驱结点的后继线索</span></span><br><span class="line">        &#123;</span><br><span class="line">            pre-&gt;rTag = <span class="number">1</span>;     <span class="comment">//前驱结点的右线索标志置1</span></span><br><span class="line">            pre-&gt;rChild = cur; <span class="comment">//前驱结点的右孩子结点指针指向当前结点</span></span><br><span class="line">        &#125;</span><br><span class="line"></span><br><span class="line">        pre = cur; <span class="comment">//更新结点指针：前驱结点指针指向当前结点</span></span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line">    <span class="keyword">return</span>;</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="comment">//主函数————————————————————</span></span><br><span class="line"><span class="function"><span class="type">int</span> <span class="title">main</span><span class="params">()</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="built_in">use_example</span>(); <span class="comment">//使用示例</span></span><br><span class="line"></span><br><span class="line">    <span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>
<hr>
<h1 id="总结"><a href="#总结" class="headerlink" title="总结"></a>总结</h1><p>不同参考资料中，二叉树和线索二叉树的描述实现各不相同，但基本思想是一致的。作者使用规范的变量命名、提供详细的步骤解析及使用示例，应用C++语言将其整合成模板，以帮助理解记忆。</p>
<hr>
<h1 id="参考资料"><a href="#参考资料" class="headerlink" title="参考资料"></a>参考资料</h1><ul>
<li>《2023版数据结构高分笔记》主编：率辉</li>
<li>《2023年计算机组成原理考研复习指导》组编：王道论坛</li>
<li>《大话数据结构》作者：程杰</li>
<li>《代码随想录》作者：孙秀洋</li>
<li><a target="_blank" rel="noopener" href="https://programmercarl.com/0102.%E4%BA%8C%E5%8F%89%E6%A0%91%E7%9A%84%E5%B1%82%E5%BA%8F%E9%81%8D%E5%8E%86.html">代码随想录 (programmercarl.com)</a></li>
</ul>
<hr>
<h1 id="作者的话"><a href="#作者的话" class="headerlink" title="作者的话"></a>作者的话</h1><ul>
<li><strong>感谢参考资料的作者&#x2F;博主</strong></li>
<li>作者：夜悊</li>
<li>版权所有，转载请注明出处，谢谢~</li>
<li><strong>如果文章对你有帮助，请点个赞或加个粉丝吧，你的支持就是作者的动力~</strong></li>
<li>文章在描述时有疑惑的地方，请留言，定会一一耐心讨论、解答</li>
<li>文章在认识上有错误的地方, 敬请批评指正</li>
<li>望读者们都能有所收获</li>
</ul>
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class="toc-text">bTree.cpp（二叉树）</span></a></li><li class="toc-item toc-level-1"><a class="toc-link" href="#tBTree-cpp%EF%BC%88%E7%BA%BF%E7%B4%A2%E4%BA%8C%E5%8F%89%E6%A0%91%EF%BC%89"><span class="toc-number">4.</span> <span class="toc-text">tBTree.cpp（线索二叉树）</span></a></li><li class="toc-item toc-level-1"><a class="toc-link" href="#%E6%80%BB%E7%BB%93"><span class="toc-number">5.</span> <span class="toc-text">总结</span></a></li><li class="toc-item toc-level-1"><a class="toc-link" href="#%E5%8F%82%E8%80%83%E8%B5%84%E6%96%99"><span class="toc-number">6.</span> <span class="toc-text">参考资料</span></a></li><li class="toc-item toc-level-1"><a class="toc-link" href="#%E4%BD%9C%E8%80%85%E7%9A%84%E8%AF%9D"><span class="toc-number">7.</span> <span class="toc-text">作者的话</span></a></li></ol></div></div></div></div></main><footer id="footer" style="background-image: url('/img/cover10.png')"><div id="footer-wrap"><div class="copyright">&copy;2023 - 2025 By 夜悊</div><div class="footer_custom_text"><a 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